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Yes. The density of the Poincare metric with respect to the spherical metric is a positive continuous function which tends to infinity at the punctures. Thus it is bounded from below by some positive …

The answer is negative. Since every neighborhood of a point on the Julia set is mapped onto the whole Julia set by some iterate of the rational function, it follows that if a small piece of the Julia …

The answer is this: $$f(z)=\int_{z_0}^z e^{Q(\zeta)}d\zeta,$$ where $Q$ is a polynomial, and this is the general form of an entire function whose Schwarzian is a polynomial. The crucial fact that $f$ …

MLC is indeed a very technical and complicated counterpart of a simple question which arises from the general theory of dynamical systems. For a generic system (in a given finitely-parametric family) …

Main areas are dynamics of automorphisms (for example, Henon maps), dynamics of endomorphisms, dynamics of foliations, and local dynamics. Eric Bedford, Tien-Cuong Dinh, John Fornaess, Misha Lyubich, …

Answer. $J(fg)$ is the full $g$-preimage of $J(gf)$. (And vice versa with interchange of $f$ and $g$). Proof. Let $A=fg$ and $B=gf$. Then we have a semi-conjugacy $gA=Bg$. Now it is a general fact, …

MR1032073 Doyle, Peter; McMullen, Curt, Solving the quintic by iteration. Acta Math. 163 (1989), no. 3-4, 151–180.

Nobody. This is the principal unsolved problem in the area, which is called MLC (That the Mandelbrot set is locally connected). Two Fields medals were awarded for partial progress in this problem. Ab …

The answer is positive and this is not difficult (a normal families argument). The boundary of the Mandelbrot set is the set of $J$-instability. Every point $c_0$ of this set is a limit of $c_n$ such …

You iterate the function $a\mapsto (\cos a)^z$ which is ill defined for complex $a$ and $z$; you need a branch cut, which is visible on some of your pictures. In holomorphic dynamics, usually entire f …

He explains his choice in lines 8-9 on p. 364 of the paper: "This metric has the curvature $-4$ for it is obtained from the hyperbolic metric by the transformation $w'=w^{1/2}$ ." Remarks. By more sop …

Your statement that iterates of the Newton method converge to a cycle almost everywhere is equivalent to the statement that for every polynomial $f$ the Julia set of the rational function $z-f(z)/f'(z …

The answer to a) is yes, and this was proved by Fatou in 1919. Sur les équations fonctionnelles Bulletin de la S. M. F., tome 48 (1920), p. 208-314. There are many generalizations of this fact. For o …

You do not tell the crucial thing: how large is $|f'(0)|$. There is no simple expression for coefficients or any other simple expression for $h$, even when $f$ is quadratic polynomial $\lambda z+z^2$. …

Yes, these are the neighborhoods of certain Misiuriewicz points, see for example the right picture in the bottom here: http://classes.yale.edu/fractals/MandelSet/MandelBoundary/Mis.html